Prove that sin(x)+sin(y)=2sin((x+y)/2)cos((x-y)/2)

We know that 1. sin(a+b) = sin(a)cos(b)+sin(b)cos(a) and 2. sin(a-b) = sin(a)cos(b)-sin(b)cos(a) Add equations 1. and 2. sin(a+b)+sin(a-b) = 2sin(a)cos(b)+sin(b)cos(a)-sin(b)cos(a) = 2sin(a)cos(b) Let x=a+b and y=a-b, hence x+y=2a so a=(x+y)/2 and x-y=2b so b=(x-y)/2 Therefoe sin(x)+sin(y) = 2sin((x+y)/2)cos((x-y)/2)